^{1}

^{2}

^{1}

^{2}

^{1}

This paper is built upon the previous developments on lateral earth pressure by providing a series of analytical expressions that may be used to evaluate vertical profiles of the effective stress and the corresponding suction stress under steady-state flow conditions. Suction stress profile is modeled for one layer sand near the ground above the water level under hydrostatic conditions. By definition, the absolute magnitude of suction stress depends on both the magnitude of the effective stress parameter and matric suction itself. Thus, by developing the Rankine’s relations in seismic state, the composing method of active and passive surfaces in sides of unbraced sheet pile is examinated and the effects of soil parameter on those surfaces are evaluated by a similar process. The relations described the quantitative evaluation of lateral earth pressure on sheet pile and the effects of unsaturated layer on bending moment and embedded depth of sheet pile in soil.

Computation of the lateral earth pressure is extremely important and it is the basic data required for the design of geotechnical structures like the retaining walls moving towards the backfill, the sheet piles, the anchors etc. The level of importance of the lateral earth pressure increases many folds under the earthquake conditions due to the devastating effects of the earthquake. Hence, to design the sheet pile both under the static and seismic conditions, the basic theory is very complex and several researchers have discussed on this topic. Initially Okabe [

Again, by using an approximate method based on a modified shear beam model Wu and Finn [

Before determining of the related formulation for evaluation of unsaturated layer effect on seismic condition, it is necessary to gain the correct profile of active and passive stresses on the unbraced sheet pile wall. Profiles of stress in soil are often used as the theoretical basis for foundation design and analysis. Establishing relationships among different stress components such as horizontal and vertical earth pressures requires stress-strain constitutive laws. The most commonly used linear stress-strain equation in elasticity is Hooke’s law. For unsaturated soil, an extended Hooke’s law in light of the suction stress concept can be derived by substituting the effective stress components in those equations.

For soil at a state of active failure, the vertical effective stress

In seismic situation, the relationship between the minimum principal stress and maximum principal stress at failure can be written as,

where k_{ae} is the coefficient of Rankine’s active earth pressure in seismic mode. Substituting Equations (1) and (2) into Equation (3) leads to,

It is evident that the first two terms on the right-hand side of Equation (4) are included in Rankine’s original theory. The third term represents the contribution of suction stress that arises in unsaturated soil. The first term induces a compressive earth pressure in the soil mass or on adjacent retaining structures. The second and third terms are tensional stresses.

If suction stress is assumed constant (i.e., the same at all depths from the ground surface) and the soil is homogeneous, the relative contribution of each term in Equation (4) can be conceptualized in

effect of these three components of stress results in a linear lateral earth pressure profile that divides the unsaturated soil zone into zones of resultant tensional stress and compressional stress. Tensional stress will act to cause soil to crack, thus nullifying the lateral earth stress within the zone near the surface.

In situations such as a soil mass located in front of a failing retaining wall or an expansive soil mass located behind a retaining wall, the horizontal earth pressure could be greater than the vertical stress induced by the overburden. Failure occurs when the horizontal stress develops to a magnitude such that the state of stress reaches the Coulombian failure stress. This general condition is referred to as the passive limit state. For soil at a state of passive failure, the vertical effective stress

The relationship between the minimum principal stress and maximum principal stress at failure can be written as,

where k_{pe} is the coefficient of Rankine’s passive seismic pressure. Substituting Equations (1) and (2) into Equation (5) leads to,

The first two terms on the right-hand side of Equation (6) fall from classical Rankine theory. The third term represents the suction stress component arising in unsaturated soil. All three terms induce compressive earth pressure.

If suction stress is assumed constant with depth and the soil is homogeneous, the relative contribution of each term in Equation (6) can be conceptualized in

In following, the distribution of pure pressure on unbraced sheet pile in granular soil is calculated at different regions and afterwards by determining of passive lengths in soil, the embedded depth will gain too.

As shown in _{1} from top will be calculated with Equation (7) as below,

where, γ is unit weight of soil around the sheet pile. Also, the quantity of _{ae} is the coefficient of active earth pressure adopting to Rankine theory for calculation of seismic earth pressure which gains from Equation (8),

where, f and δ respectively are the friction angle of soil and the angle of wall friction, and θ is the inertial angle of earthquake which is defined as,

In this manner, the active earth pressure at z = L_{1} + L_{2} will be gained in form of Equation (10),

where, γ' is the effective unit weight of soil that equals to γ' = γ_{sat} − γ_{w}. The active earth pressure at turning point, E, placed in depth of z from top of sheet pile, can achieve be Equation (11),

The passive lateral earth pressure at depth z > L_{1} + L_{2} will be achieved from Equation (12),

where, k_{pe} is the coefficient of passive earth pressure by Rankine criterion which be calculated by Equation (13),

with combining of Equations (11) and (12), the pure lateral earth pressure at ground level becomes,

then, the location of point E under the ground level, may be expressed by L_{3} for zero pressure on sheet pile in Equation (15) as,

which the value of p_{2} presents in Equation (9). With using Equation (6), it is considered that the gradient of pressure graph DEF is equal with 1 vertical to γ'(k_{pe} − k_{ae}) horizontal. Thus, in this graph, the value of p_{3} may be written as,

then, the passive earth pressure p_{p} from right to left direction at z = L + D and the active pressure p_{a} with opposite direction corresponding to this level will be expressed as below forms,

At last, the pure lateral earth pressure on the end of the sheet pile will be equal to Equation (19),

which by considering of D = L_{3} + L_{4} in Equation (19), the value of p_{5} may be written as Equation (20),

with writing of horizontal forces equilibrium for unit of sheet pile length, Equation (21) may be gained as,

where P is the surface of under pressure graph ACDE. Also from Equation (22), the length of L_{5} can expressed as following,

Furthermore, by using of moment balance around the sheet pile leg, the Equation (23) will be achieved,

where, the value of

where the coefficients of Ai will be expressed as below,

Now, with solving this achieved equation at Equation (23), the value of L_{4} will be gotten in basis of geometric sheet pile conditions and resistant characteristics of soil. Thus, the embedded depth of unbraced sheet pile in soil can be expressed by the sum of D = L_{3} + L_{4}. In following, the value of maximum bending moment on sheet pile beneath of turning point E, will be gotten. Thus, to this purpose, it must be found the zero shear point on pressure graph. As shown in

The place of zero shear point will be gained by setting of the value P equal to zero, leads to the following expression,

with determining of zero shear point, the value of maximum bending moment on sheet pile becomes,

Ultimately, for the better evaluation form lateral earth pressure of sheet pile and the embedded length in granular soil, basis of gained relations in above, in this section, several programs is prepared to parametric study more conveniently.

A sandy unsaturated soil deposit backfilled the unbraced sheet pile near the sea water which the upper layer of it may be considered as a unsaturated soil above the water level. The soil has the following properties: internal friction angle f = 30^{°}, no cohesion, unit weight 16 kN/m^{3}, and suction stress 20 kPa. Also, it is supposed that there is an earthquake with 0.2 g for kh horizontal seismic coefficient. These values are constant throughout the entire depth of the unsaturated zone.

Thus, with presenting of characteristics in

The effect of combining of suction stress results in a linear lateral earth pressure profile that divides the unsaturated soil zone into zones of resultant tensional stress and compressional stress. Tensional stress will act to cause soil to crack, thus nullifying the lateral earth stress within the zone near the surface.

Quantity | Value |
---|---|

unit weight of backfill soil (γ) | 16 kN/m^{3} |

saturated unit weight of soil (γ_{sat}) | 19 kN/m^{3} |

internal soil friction angle (f) | 30˚ |

cohesion constant (c) | 0 |

angle of wall friction (δ) | 0.67f |

suction stress parameter (χ) | 0.5 |

suction stresses (u_{a} − u_{w}) | 10 - 50 kPa |

According to

Similar to the numerical example in previous section, by using of equations in previous part, the lateral earth pressure on unbraced sheet pile may be calculated at any depth conveniently and the distribution graph can be drawn too. Hence, in computing relations, the geomechanical characteristics present in

According to inserted specifications in ^{3} and the saturated unit weight of 19 kN/m^{3}. So, for investigating of internal soil friction angle on lateral pressure distribution, the amount of it can be changed between 25 up to 45 degrees for rigid soil type. Thus, with considering to this object, that states the angle of wall friction is a function of internal soil friction angle (δ = 0.67f), the value of δ will be changed in presented relations too. Of course, the unbraced sheet pile height is divided in two constant lengths L1 = 2 m and L2 = 3 m outside of soil, and two passive lengths L3 and L4 inside of granular soil. Further- more, the seismic condition of problem may be applied by the horizontal seismic coefficient of kh equal to 0.2 g. Hence, the distribution of lateral earth pressure and the embedded depth of sheet pile in soil, corresponding to maximum influencing lateral pressure in seismic condition, may be drawn in

As shown in

friction angle is so noneconomical and unaccomplishment. So, it is reasonable that for stabilizing of backfilled soil at seismic conditions, is used some cluster piles for sheet pile or changing the soil around it with the better resistant specifications. Also according to

The effect of variation in suction parameter on pressure distribution for unbraced sheet pile can be achieved by studying of

Further the researcher has also analyzed that the presence of water in the backfill may increase the seismic pressure. In about this case, for unsaturated earthfill, the saturated unit weight of the soil shall be adopted while calculating the seismic

active earth pressure increment or passive earth pressure decrement using the Equation (4) and (6) as discussed in preceding pages. Now in this section, the lateral earth pressure is shown by variation of the matric suction stress in unsaturated soil layer of backfilled soil in

As you seen in the previous figures, the embedded depth of unbraced sheet pile in granular soil depends on the free length of it. Hence, with normalizing of the lengthes, the variations of embedded depth will be more conveniently studied by changing in the other parameters values and also the effects of free length value can be omitted. Thus, as it shown in

Now, we can consider to a numerical example of determination of embedded depth by using of

According to past relations, the embedded depth of unbraced sheet pile in granular soil depends on the matric suction, the effective suction parameter and the internal soil friction. Thus, as it shown in

According to

In general, by generating and using the above figures such as Figures 8-10, the embedded depth of sheet piles in granular soils with unsaturated situation can get without any calculations, which will be used profitably for designer engineers.

According to the past equations, the value of maximum bending moment on

sheet pile will be gotten beneath of the turning point E. Thus, with determining of zero shear point, the value of maximum bending moment on sheet pile will be found. Thus, by considering to the depending of maximum bending moment on active and passive lateral coefficients, it is founded that the variating in internal soil friction can be important. And also, the changing in some other parameters such as suction stress and effective suction parameter will be affects on the maxi- mum bending moment value with depending of the place of zero shear point to them. Hence, as shown in

Now, by increasing in soil resistant specification, the maximum moment may be decreased. Thus, for a constant situation for unsaturated upper soil layer that is shown in

In present paper, the unbraced sheet pile behavior of lateral earth pressure and embedded depth is studied by changing in saturation condition. Thus, by variating the matric suction stress and the effective suction parameter, the distribution of lateral earth pressure is investigated by some graphs which explained the effects of any variation on them. And also in several graphs, the effects of soil mechanical parameters are obvious on the embedded depth of unbraced sheet pile in granular around soil. Of course, this matter is presented by normalizing

of the embedded depth with the free length of sheet pile. And in following, the maximum lateral earth pressure on sheet pile is considered in each graph from the first of paper and justified the reasons of variations of it by changing of embedded length. At last, the maximum bending moment of soil on unbraced sheet pile is considered with different internal soil friction angles in a normal curve. Thus, at a general state, it is cleared that the variations of saturating conditions in seismic manner are affected on lateral earth pressure and the embedded depth of unbraced sheet pile in different forms. Hence, it is so important that the designers may consider the effects of unsaturated soil layer on unbraced sheet pile behavior.

Jahangir, M.H., Soleymani, H. and Sadeghi, S. (2017) Evaluation of Unsaturated Layer Effect on Seismic Analysis of Unbraced Sheet Pile Wall. Open Journal of Marine Science, 7, 300-316. https://doi.org/10.4236/ojms.2017.72022